ps5 - Problem Set 5 not to be turned in Problem 1 An...

Problem Set 5 - not to be turned in Problem 1 An experiment consists of drawing balls without replacement from an urn containing 10 balls numbered 1 , 2 ,..., 10. For each of the following statements, answer True if the statement is always true, and answer False otherwise. If you answered True, provide a brief argument (e.g., you can cite a theorem proved in class or in a book.). If you answered False, provide the correct answer. It is ﬁne to leave answers containing factorials, combinations and permutations as they are without evaluating them. (a) For E,F ∈ F , P( E ∩ F c ) = P( E )-P( E ∩ F ). (b) Let E n be the event that ball number 1 is still in the urn after the n th draw, 1 6 n 6 10. Then, E n is an increasing sequence of events. (c) The total number of diﬀerent orders in which these 10 balls can be drawn is 10 10 . (d) The probability that we get ball number 1 in the very ﬁrst draw is 1 10 . (e) The probability that we get ball number 1 in exactly the fourth draw is 1 10 . (f) After four balls are drawn, the probability that the minimum numbered ball being the fourth is 1 / ( 10 4 ) . (g) Suppose we distribute these balls among three diﬀerent urns, urn 1, urn 2 and urn 3. There are ( 10 3 ) possible diﬀerent arrangements. (h) Suppose each ball is placed in one of the three urns, urn 1, urn 2 or urn 3 randomly with

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